Both naught and nought mean nothing, and in American English they are more or less interchangeable (though naught is the more common spelling). Elsewhere, they are different. Nought is conventionally used in British English for the number zero—for example:
This season that figure is at 1.36: they are fifth despite scoring just 25 goals and having a goal difference of nought. [Guardian]
The number opting for the subject ranges from nought to 90 per cent. [The Independent]
In both British English and American English, naught is used in nonmathematical contexts to mean nothing, usually in the phrases for naught and come to naught—for example:
But all this work could be for naught if the NFL owners and players fail to agree on a new collective bargaining agreement by March 3. [STL Today]
My luck went for naught a week or two later when the report was published about the dangerously high levels of mercury in Lake Erie fish. [Casper Star-Tribune]
They don’t call it Beantown for naught. [Huffington Post]
In these examples, for nothing would convey the same meaning as for naught.